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IN   MEMORIAM 
FLORIAN  CAJORl 


PR 


LEMENTARY    ALGEBRA. 


*'S.„    : 
1.30       ! 


BY 


H.   W,   KEIGWIN. 


o**to 


BOSTON: 

PUBLISHED    BY  )MPANY. 

1886. 


PEIi^CIPLES 


^^ 


Elementary   Algebra. 


BT 

H.  W.J^IGWIN. 


o}«<o 


BOSTON: 

PUBLISHED   BY   GINN   &  COMPANY. 

1886. 


CAJORI 


Entered,  according  to  Act  of  Congress,  in  the  year  1886,  by 

H.  W.  KEIGWIJi, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


J.  8.  CusHiNG  &  Co.,  Printers,  Boston. 


QI53 


NOTE. 


This  little  boob  is  intended  as  an  outline  of  thorongli  oral 
instruction,  and  is  all  the  ''text"  I  have  found  necessary 
to  put  into  my  pupils'  hands.  It  should  of  course  be  accom- 
panied by  a  good  set  of  exercises  and  problems. 

Pupils  study  algebra  with  much  more  interest  and  profit 
when  they  are  led  to  discover  and  to  interpret  their  own 
formulas  and  to  compose  their  own  rules.  I  have  left  much 
for  the  teacher  and  the  pupil  to  do,  and  have  aimed  to  make 
the  outline  brief,  accurate,  and  useful  as  a  text-book. 

Matawaf,  New  Jersey, 
October,  1886. 


I  y 


PEII^OIPLES 


ELEMEISTTARY  ALGEBRA. 


CHAPTER  I. 

DEFINITIONS,   ETC. 

1.  The  numbers  of  Algebra  extend  from  zero  in  two  oppo- 
site directions ;  those  in  one  direction  are  called  positive, 
those  in  the  opposite  direction  negative. 

'S.  Positive  and  negative  numbers  are  distinguished  hj 
prefixing  to  a  positive  number  the  sign  +»  ^^^  to  a  negative 
number  the  sign  — . 

The  sign  +  is  often  omitted  when  it  can  be  readily  under- 
stood. 

3.  The  symbols  of  algebraic  numbers  are  the  figures  of 
arithmetic  and  the  letters  of  the  alphabet.  Thus,  2,-7,  n, 
—  y,  D,  a  denote  algebraic  numbers. 

4.  The  signs  +,  — ,  X,  -^  have  the  same  general  meaning 
as  in  arithmetic. 

The  X  is  little  used,  multiplication  being  indicated  by  writ- 
ing the  factors  in  line.  When  the  factors  are  the  numbers 
of  arithmetic  they  are  sometimes  separated  by  a  point.    Thus  : 


4  PEINCIPLES   OF    ELEMENTARY   ALGEBRA. 

axhxm  is  written  aim ;  3  X  5  is  written  3*5;  3  X  a  + 
3x5Xa  +  3x5x7xa  is  written  3a  + 3  '  5(2  + 3  •  5  •  7a. 
Division  is  generally  indicated  by  writing  the  dividend  as 
the  numerator,  and  the  divisor  as  the  denominator  of  a  frac- 
tion. 

6.  The  exponential  notation  is  the  same  in  algebra  as  in 
arithmetic.  Thus  :  2^  means  2  •  2  *  2  ;  y^  means  yyyy.  When 
a  factor  occurs  with  no  written  exponent,  the  exponent  1  is 
understood. 

6.  The  radical  sign  -yj  is  used  with  the  same  meaning  in 
algebra  as  in  arithmetic. 

7.  The  sign  of  aggregation  indicates  that  the  numbers 
enclosed  by  the  sign  are  to  be  taken  collectively.  The  signs 
used  are  the  parenthesis  marks  ( ),  brackets  [  ],  braces  \\, 

and  vinculum or  bar  |.     Thus  :  Va^  +  b"^  means  that  the 

sum  of  o?  and  h^  is  first  to  be  found,  and  then  the  square  root 
of  that  sum  is  to  be  found. 

+  x 

—  y   is  equal  to   ?>(x  —  y-\-  z), 

+  z 

8.  Any  collection  of  algebraic  symbols  of  number  with  any 
of  the  signs  just  described  is  called  an  algebraic  expression^ 
or  an  expression^  or  a  quantity. 

9.  The  parts  of  an  expression  that  are  connected  by  + 
and  —  signs  are  called  terms. 

If  a  term  contain  no  letter  it  is  called  a  numerical  term. 

10.  If  an  expression  consist  of  a  single  term  it  is  called 
a  monomial ;  if  of  two  terms,  it  is  called  a  binomial ;  if  of 
three  terms,  a  trinomial.  Any  expression  of  more  than  one 
term  is  often  called  a  polynomial. 


DEFINITIONS,    ETC.  6 

11.  A  coefficient  is  a  multiplier  of  any  quantity.  When 
several  quantities  are  multiplied  together  the  product  of  any 
of  them  may  be  considered  the  coefficient  of  the  remaining 
product.  Thus,  in  3  alp-x^  3  is  the  coefficient  of  aVx  ;  3  aV  is 
the  coefficient  of  ^  ;  3  ao;  is  the  coefficient  of  6^ 

When  the  term  is  negative,  the  coefficient  is  generally  sup- 
posed to  involve  the  sign.  Thus,  in  —bVx  we  should  gen- 
erally say  that  —  5  is  the  coefficient  of  b'^x. 

When  a  term  occurs  with  no  numerical  coefficient,  the 
coefficient  1  is  understood. 

12.  Like  terms  or  similar  terms  are  such  as  have  the  literal 
part  the  same.     Thus:  abx,  babx,  —  11  aSo;  are  like. 

13.  The  degree  of  a  term  is  equal  to  the  number  of  literal 
factors  it  contains,  or  it  is  equal  to  the  sum  of  the  exponents 
of  the  letters. 

14.  The  numerical  value  of  an  algebraic  quantity  is  its 
value  as  an  arithmetical  number.     It  may  be  either  +  or  — . 

15.  The  absolute  value  of  an  algebraic  quantity  is  its  value 
independent  of  its  sign.  Thus,  in  the  expression  a^  —  8a5  +  6^ 
if  a  is  2  and  J  is  1,  the  numerical  value  of  the  expression  is 
— 11,  and  its  absolute  value  is  11. 

16.  The  degree  of  an  expression  is  the  degree  of  the  highest 
term  in  the  expression. 

17.  The  sign  of  equality  =  is  used  with  the  same  meaning 
as  in  arithmetic. 


CHAPTER  11. 

ADDITION  AND   SUBTRACTION. 

18.  Negative  numbers  are  counted  in  a  direction  opposite 
to  the  direction  of  positive  numbers. 

19.  Positive  numbers  are  added  as  in  arithmetic.  When 
two  or  more  negative  numbers  are  added,  the  absolute  value 
of  the  sum  is  the  same  as  if  they  were  positive  numbers,  but 
the  sum  is  preceded  by  the  sign  — . 

20.  When  a  negative  number  is  added  to  a  positive,  the 
result  is  found  by  counting  the  positive  number  forward  (that 
is,  increasingly),  and  then  counting  the  negative  number  back- 
ward. The  last  number  counted  is  called  the  sum.  Thus,  to 
add  4  and  —  6.  Count  one,  two,  three,  four  ;  then  three,  two, 
one,  zero,  minus  one,  minus  two  ;  4  —  6  =  —  2.  To  add  m 
and  —  n ;  count  m  units,  then  count  n  units  in  the  opposite 
direction.  The  last  number  counted  is  the  sum  of  m  and  —  n, 
or  it  is  rrh-\-  (—  n). 

21.  Subtraction  may  be  considered  as  a  counting  backward, 
and  if  we  subtract  n  units  from  m  units,  we  count  m  units ; 
then  we  count  in  the  opposite  direction  n  units.  The  last 
number  counted  is  the  difference  of  mi  and  n,  or  it  is  m  —  (^-\-7i). 

This  operation  is  the  same  as  the  one  just  described  (20), 
and  plainly  the  result  7?i  —  (+  '^'^  is  the  same  as  m  +  (—  ^)- 
This  means  that  the  addition  of  a  negative  number  is  the  same 
as  the  subtraction  of  an  equal  positive  number.  Both  these 
expressions  m  +  (—  n)  and  w  —  (+  n)  are  generally  written 
m  —  n. 


ADDITION   AND    SUBTRACTION.  7 

22.  Let  n  denote  any  number.  If  we  add  a  certain  num- 
ber of  Ti's,  say  a  ns,  to  some  other  number  of  ns,  say  b  7i*s, 
we  may  indicate  it  thus  :  an  +  bn.  It  is  plain  that  this  sum 
an  +  bn   must  contain  n  just  a~\-b  times  ;  that  is, 

an -\- bn  =  (a -{-  b)  n. 

If  a  and  b  are  arithmetical  numbers,  an  and  bn  are  similar 
terms.     So  we  get  a  rule  for  addition  : 

Add  the  coefficients  of  similar  terms,  and  prefix  the  sum  to 
the  common  letter. 

Ex.  2a-5b  +  Sc+7a  +  9b-~Sc 

-  (2  +  7)  a  +  (-  5  +  9)  &  +  (3  -  8)  c 
=  9a  +  4:b-5c, 

23.  Any  expression  or  quantity  means  that  the  terms  are 
to  be  combined  according  to  the  principles  of  addition.  If 
there  occur  a  quantity  in  a  parenthesis  to  be  added,  it  is  plain 
we  may  remove  the  sign  of  aggregation  and  add  the  terms  at 
once. 

24.  Suppose  now  we  have  to  subtract  a  binomial.  Suppose 
the  expression  -^  _  C^  _i_  i)\ 

If  from  m  we  subtract  one  term  at  a  time,  when  we  have  sub- 
tracted a  (and  get  m  —  a),  we  have  not  taken  away  enough ; 
we  were  to  subtract  a  and  b,  so  there  remains  b  to  be  sub- 
tracted. If,  then,  from  m  we  take  a  and  b  one  after  the  other 
(that  is,  if  we  remove  the  parenthesis  and  subtract)  we  shall 
get  771  —  a  —  b.    It  follows  that 

m  —  (a  -}-  b)  =^  m  —  a  ~  b. 

In  the  first  expression  the  signs  of  both  a  and  b  are  + ;  in 
the  last  expression  they  are  both  — ,  evidently  changed  by 
removing  a  parenthesis  when  it  is  preceded  by  a  minus  sign. 

Again,  suppose  the  expression 

7n  —  (a  —  b). 


8  PRINCIPLES   OF   ELEMENTARY   ALGEBRA. 

As  before,  let  ns  subtract  a  term  at  a  time.  When  we  have 
taken  a  from  ra  we  have  taken  too  much,  for  we  are  required 
to  take  a  —  h  (that  is, , something  less  than  a).  Plainly  m  —  a 
is  a  result  too  small  by  J,  and  we  must  add  6.     Then  we  have 

m  — a  +  J. 

Therefore,  m-~{a—h)  =  m  —  a  +  b, 

and  in  this  case  we  have  changed  the  signs  of  both  terms,  by 
removing  a  parenthesis  when  preceded  by  a  minus  sign. 

25.  Of  course  this  principle  can  be  extended  to  any  number 
of  terms,  and  by  a  similar  principle  aggregation  signs  can  be 
introduced  into  any  polynomial  expression. 

26.  From  this  principle  for  removing  the  parenthesis  sign, 
we  get  the  rule  for  subtraction : 

Change  the  signs  of  the  terms  to  be  subtracted,  and  proceed 
as  in  addition. 

27.  Often  many  or  all  of  the  signs  can  be  changed  mentally, 
and  the  rewriting  of  the  whole  expression  can  be  avoided. 

28.  It  follows  from  the  last  part  of  24  that, 

m  —  (—  J)  =  m  +  5. 

This  is  true  whatever  m  may  be,  and  if  m  —  0  it  is  still  true 
that, 


CHAPTEK  III. 

MULTIPLICATION   AND   DIVISION. 

MULTIPLICATION. 

29.  We  have  seen  (4)  that  multiplication  is  indicated  by 
writing  the  factors  successively  without  the  sign  X. 

Thus:  lab  means  7  times  a  times  b\  (a-~?>b){x-{-2y~\-\V) 
means  the  product  of  the  binomial  into  the  trinomial. 

We  have  seen,  too  (22),  that  the  product  of  a  monomial 
into  a  polynomial  is  found  by  multiplying  every  term  of  the 
polynomial  by  the  monomial,  and  connecting  the  results. 

Thus  :  a  {bx  +  ac)  =  abx  +  o?c. 

Evidently  if  we  have  several  terms  in  each  factor  we  can 
get  the  true  product  by  combining  the  partial  products.  We 
-have  only  to  consider  the  law  of  signs, 

30.  Let  the  absolute  values  of  two  numbers  be  denoted  by 
m  and  n. 

(1)  If  both  are  positive,  the  product  is  positive. 

(2)  If  one,  say  m^  is  positive,  and  the  other,  n,  is  negative, 
we  know  that  if  we  add  any  number  of  —  n's,  the  result  will 
be  negative ;  if  we  take  m  such  numbers,  the  product  will  be 
—  mn ;  that  is, 

(+  m)  (—n)  =  —  mn. 

(3)  It  is  assumed  in  algebra  that  the  order  of  the  factors 
makes  no  difference  in  the  value  of  the  product.  Therefore, 
(+  m)  (—  n)  =  (—  n)  (-{-m)  =  —  wm.     As  (—  n)  (+  m)  is  neg- 


10  PEINCIPLES   OF   ELEMENTARY   ALGEBRA. 

ative,  (—  m)  (+  n)  will  also  be  negative  ;  for  the  values  of  m 
and  n  are  general.     Therefore, 

(—  7)%)  (+  ^)  =  —  mn. 

(4)    If  both  are  negative, 

(—  7)1)  {—7i)^=  —  7n  ( —  n)  ; 
writing  the  coefficient  of  —  m,    =  —  1 7?^  (—  n)  ; 
from  (3)  above,  first  part,  =  —  1  (—  n)  m  ; 

dropping  the  1,  =  —  (—  n)  m ; 

by  28,  =  +  n(m); 

^=-{-77171. 

Therefore,  (—  m)  (— -  7i)  =  +  "rriTi. 

31.  We  infer  :  If  both  the  signs  are  plus,  or  if  both  are 
minus,  the  product  is  plus.  If  either  one  is  minus,  the 
product  is  minus. 

32.  By  arranging  the  signs  in  pairs,  the  sign  of  the  product 
of  any  number  of  factors,  can  be  determined,  and  it  will  be 
seen  that : 

Any  number  of  plus  signs  gives  plus. 
'  An  even  number  of  minus  signs  gives  plus. 
An  odd  number  of  minus  signs  gives  minus. 

33.  Eule  for  multiplication :  Multiply  every  term  of  one 
polynomial  by  every  term  of  the  other  polynomial,  and  com- 
bine the  partial  products. 


MULTIPLICATION   AND    DIVISION.  11 


DIVISION. 

34.  In  Division  we  have  given  a  product  and  one  factor  to 
find  the  other  factor. 

The  process  is  similar  to  division  in  arithmetic,  and  the  law 
of  signs  is  similar  to  the  law  in  31,  as  will  be  seen  by  consid- 
ering the  cases  in  30. 

35.  Care  should  be  taken  to  arrange  the  terms  of  both  divi- 
dend and  divisor  according  to  the  progressive  powers  of  the 
same  letter.  This  affords  a  more  satisfactory  form  in  the 
answer. 


CHAPTER  IV. 

GREATEST  COMMON   DIVISOR  AND   LEAST 
COMMON   MULTIPLE. 

GREATEST   COMMON   DIVISOR. 

36.  The  greatest  common  divisor  of  two  or  more  quantities 
is  the  quantity  of  highest  degree  that  will  divide  every  one 
of  them. 

The  greatest  common  divisor  is  denoted  by  the  abbreviation 
G.C.D.  It  is  also  called  the  greatest  common  measure  (abbre- 
viation, G.C.M.)  and  the  highest  common  factor  (abbreviation, 
H.C.F.). 

37.  When  the  quantities  whose  G.C.D.  is  to  be  found  are 
monomials,  it  can  best  be  found  by  inspecting  the  quantities 
and  determining  their  factors.  The  product  of  the  factors 
common  to  all  is  the  G.C.D. 

38.  "When  the  quantities  are  polynomials,  we  find  the  G.C.D. 
of  any  two,  then  of  this  divisor  and  a  third  polynomial,  and 
so  on. 

39.  The  method  followed  is  similar  to  the  method  in 
arithmetic  : 

Divide  the  polynomial  of  higher  degree  by  the  other  poly- 
nomial. 

Divide  the  divisor  by  the  last  remainder,  and  so  on  till 
there  is  no  remainder. 

The  last  divisor  is  the  G.C.D. 

40.  To  avoid  fractional  coefficients^  this  rule  is  slightly 
modified : 


LEAST   COMMON   MULTIPLE.  .   13 

(1)  Remove  all  monomial  factors  from  the  polynomials,  and 
save  any  common  factors  as  a  part  of  the  G.C.D. 

(2)  Remove  any  factor  from  any  expression  in  the  course 
of  the  work  when  it  will  facilitate  the  work. 

(3)  Whenever  the  first  term  of  a  divisor  is  not  contained 
an  integral  number  of  times  in  the  first  term  of  its  dividend, 
introduce  any  required  factor. 

This  rule  will  be  proved  later. 

LEAST  COMMON   MULTIPLE. 

4L  The  least  common  multiple  of  two  or  more  quantities 
is  the  quantity  of  lowest  degree  that  is  divisible  by  every  one 
of  them. 

The  least  common  multiple,  also  called  the  least  common 
measure,  is  denoted  by  the  abbreviation  L.G.M. 

42.  The  L.G.M.  of  several  quantities  is  plainly  the  product 
of  the  factors  occurring  in  the  quantities,  each  factor  taken 
just  times  enough  so  that  the  multiple  will  contain  every  one 
of  the  quantities, 

43.  When  the  quantities  cannot  be  factored  by  inspection, 
the  L.O.M.  of  two  quantities  is  found,  then  of  this  multiple 
and  a  third  quantity,  and  so  on, 

44.  The  L.O.M.  of  two  polynomials  is  found  by  multiplying 
together  the  G-.O.D.  of  the  polynomials  and  the  remaining 
factor  in  each  polynomial. 

The  product  of  the  three  factors  is  the  L.O.M.  of  the  two 
polynomials. 


CHAPTER  V. 

FORMULAS  AND   FRACTIONS. 

FORMULAS. 

45.  A  formula  is  an  algebraic  expression  which,  from  its 
frequent  application,  is  of  special  use.  The  following  are 
worth  memorizing : 

(1)  {a  +  by  =  a^^2ah  +  b\ 

(2)  a?-h''={a-^b){a-b). 

(3)  a?  +  F={a  +  b){a^-ab  +  b% 

(4)  a^-F^ia-bXa'  +  ab  +  b"). 

(5)  a*-b*  =  {a'-b'){a'  +  b'), 

=  (a-bXa  +  b)(a'  +  P), 

=  (a  -  b){a'  +  d'b  +  ab'  +  5'). . 

(6)  a-'  +  ¥  =  (a  +  b){a*  -  a'b  +  aV  -  aW  +  5*). 

(7)  a^  -  6'  =  (a  -  6)(a*  +  a?b  +  d'b'  +  ab'  +  b% 

(8)  a^-b^  =  {a^-b'){a?  +  b% 

=  {a-b){a^  +  ah  +  b'){a+b){a}-ab  +  l''); 
also  =^(a?~-¥){a*  +  a^¥  +  b'). 

(9)  {x+a)(x+b)  =  x'-\-{a  +  b)x  +  ab. 

FRACTIONS. 

46.  The  laws  which  govern  the  treatment  of  fractions  is  In 
general  the  same  as  in  arithmetic.     These  cautions  are  useful : 


FOEMULAS   AND    FRACTIONS.  16 

(1)  The  sign  which  precedes  the  fraction,  called  the  sign 
of  the  fraction,  may  be  regarded  as  belonging  to  either  the 
numerator  or  the  denominator. 

(2)  The  numerator  and  the  denominator  are  regarded  as 
''  wholes,"  and  are  to  be  treated  as  if  enclosed  in  parentheses. 

This  is  especially  to  be  considered  when  a  factor  precedes 
or  follows  the  fraction,  or  when  the  fraction's  sign  is  minus. 

(3)  Changing  the  sign  of  either  the  numerator  or  the  denom- 
inator changes  the  sign  of  the  fraction  ;  and 

Changing  the  signs  of  both  numerator  and  denominator  does 
not  change  the  sign  of  the  fraction. 

47.    Illustrations : 

(1)  a'-ah  +  b'_-  (a'-ab  +  b'')  _a'-ab  +  b^ 

TYl  —  X  VI  —  X  —  {m  —  x^ 

-a^^ab-  b''      o}~ab  +  V 


m  —  X  x  —  m 


In  this  illustration,  in  all  the  forms  after  the  first,  the  sign 
of  the  fraction  (being  unwritten)  is  plus. 

/o\       a  +  &    {a-\-b)x ax-{-bx a-\-b 


CHAPTER  VI. 

EQUATIONS  AND   THE   SOLUTION   OF   SIMPLE 
EQUATIONS. 

48.  An  equation  is  an  algebraic  statement  that  two  quan- 
tities are  equal. 

The  expression  on  the  left  of  the  sign  of  equality  is  called 
the  first  member;  that  on  the  right  is  called  the  second 
member. 

49.  An  identical  equation  (or  an  identity)  is  an  equation 
which  contains  only  numerical  terms ;  or,  it  is  one  which  is 
true  whatever  values  be  assigned  to  the  letters.  The  illustra- 
tions and  formulas  in  Chapter  V.  are  instances  of  identities. 

50.  An  equation  of  condition  is  one  which  is  true  only  for 
certain  values  of  a  letter  which  represents  the  unhnown 
quantity.  These  values  are  called  the  roots  of  the  equation, 
and  finding  them  is  called  solving  the  equation. 

When  a  root  is  substituted  for  the  unknown  quantity,  the 
equation  of  condition  becomes  an  identity. 

By  *'  equation  "  is  meant  an  equation  of  condition,  unless 
otherwise  stated  or  implied. 

51.  Known  quantities  are  those  whose  values,  it  is  assumed, 
are  given. 

52.  Unknown  quantities  are  those  whose  values  are  to  be 
found.  The  final  letters  of  the  alphabet  are  usually  reserved 
for  these,  though  any  letters  may  be  used. 


SOLUTION   OF   SIMPLE   EQUATIONS.  17 

53.  The  degree  of  an  equation  is  indicated  by  tlie  largest 
number  of  unknown  factors  which  occurs  in  any  term. 

Thus:  ^x^  +  ay  +  2x'y-=-0 

is  of  the  third  degree,  determined  by  the  three  unknown  fac- 
tors in  the  term  2x^y.  The  equation  should  be  free  from 
parentheses,  fractions,  and  radical  signs,  so  far  as  the  unknown 
quantities  are  concerned,  before  its  degree  is  determined. 

54.  The  following  axioms  (or  assumed  truths)  are  useful  in 
transforming  equations : 

(1)  Quantities  equal  to  the  same  quantity  are  equal  to 
each  other. 

(2)  If  equal  quantities  be  added  to  (or  subtracted  from) 
equal  quantities,  the  results  are  equal. 

(3)  If  equal  quantities  be  multiplied  (or  divided)  by  equal 
quantities,  the  results  are  equal. 

(4)  If  equal  powers  (or  equal  roots)  of  equal  quantities  be 
taken,  the  results  are  equal. 

Axioms  (2),  (3),  and  (4)  may  be  summed  up  in  : 

Similar  operations  upon  equal  quantities  give  equal  results. 

55.  An  equation  is  not  a  quantity,  and  it  cannot  be  multi- 
plied or  otherwise  treated  as  a  quantity ;  but  we  sometimes 
speak  of  multiplying  an  equation,  etc.,  meaning  thereby  mul- 
tiplying both  members  of  the  equation,  etc. 

56.  Any  term  may  be  transposed  from  one  side  of  an  equa- 
tion to  the  other  by  changing  its  sign.  This  follows  from 
axiom  (2).     Thus : 

a-{-b  —  c^=  X  (1) 

—  a  =     —  a  identity,  (2) 

adding  member  to  member,      b  ~  c^=  x ~  a  (3) 


18  PRINCIPLES    OF   ELEMENTARY   ALGEBRA. 

and  a  has  been  transposed  from  the  first  to  the  second  member 
of  (1),  and  its  sign  is  changed. 

If  we  transpose  all  the  terms  of  (1),  we  get  .    • 

—  x^=  —  a  —  h-\-c\  (4) 

therefore,  we  may  change  the  sign  of  every  term  in  an  equation 
without  destroying  the  equality. 

57.  Any  denominator  may  be  removed  from  an  equation 
by  use  of  axiom  (3) ;  and  all  the  denominators  may  be 
removed,  or  the  equation  may  be  "  cleared  of  fractions,"  by 
multiplying  the  equation  through  by  a  multiple  of  the 
denominators. 

58.  To  solve  an  equation  of  the  first  degree  with  one 
unknown  quantity  (often  called  a  simple  equation)  : 

Clear  of  fractions. 

Bring  all  the  terms  containing  the  unknown  quantity  to 
the  first  member,  and  all  the  other  terms  to  the  second 
member. 

Divide  both  members  by  the  coefiicient  of  the  unknown 
quantity. 


Ex. 

|  +  f  =  m.+  ^; 

(1) 

clearing, 

lx  +  2a  =  2bmx-^2bl; 

(2) 

transposing, 

bx  —  2  bmx  =  2bl—2a; 

(3) 

factoring, 

(b  —  2bm)x  =  2bl~2a; 

(4) 

dividing, 

2bl-2a 

(5) 

'■l^^,      («) 


CHAPTEE  VII. 

FIRST  DEGREE   EQUATIONS   CONTAINING  TWO 
OR   MORE   UNKNO^sATN   QUANTITIES. 

.59.  If  we  have  an  equation  containing  two  unknown  quan- 
tities, it  is  called  indeterminate.  It  will  be  seen  upon  trial 
that  an  indefinite  number  of  pairs  of  values  for  the  unknown 
quantities  can  be  found. 

60.  If  we  have  two  first  degree  equations  containing  two 
unknowns,  the  equations  can  in  general  be  solved  ;  that  is, 
such  a  pair  of  values  for  the  unknowns  can  be  found  that  both 
equations  will  be  satisfied. 

61.  Elimination  is  the  process  of  combining  two  or  more 
equations  in  such  a  way  as  to  remove  one  or  more  of  the 
unknown  quantities.     The  three  common  methods  are  : 

I.    By  addition  or  subtraction. 
II.    By  comparison. 
III.    By  substitution. 

I.    By  addition  or  subtraction. 

Transform  one  or  both  the  equations  so  that  the  coefficients 
of  one  of  the  unknowns  shall  be  absolutely  equal. 

Add  or  subtract  the  equations  according  as  the  equal  coeffi- 
cients have  opposite  or  like  signs. 

This  method  is  the  one  most  commonly  used. 


20  PEINCIPLES   OF    ELEMENTARY    ALGEBKA. 

II.  By  comparison. 

Write  the  equations  in  the  form : 

x  =  ay^rh,  (1) 

X  =  my  +  7t ;  (2) 

then  form  a  new  equation  by  writing  the  second  members 
equal : 

ay  +  5  —  TYiy  +  n,  (3) 

This  method  is  often  used  when  the  known  quantities  are 
literal. 

III.  By  substitution. 

Write  one  of  the  equations  in  the  form : 

x  =  ay'\-h, 

and  substitute  the  second  member  in  place  of  x  in  the  other 
equation. 

This  method  is  often  used  when  one  of  the  equations  is  of 
the  second  degree. 

62.  It  is  evident  that  this  process  may  be  extended  to  three 
or  more  first  degree  equations,  containing  three  or  more  un- 
known quantities. 

63.  From  the  final  equation  containing  only  one  unknown 
quantity,  we  may  get  its  value  (by  Ch.  VI.),  and  by  succes- 
sive substitutions  the  values  of  the  other  unknowns  may  be 
found. 

64.  Equations  which  can  be  combined  for  elimination  are 
called  simultaneous  equations. 

65.  Equations  which  can  be  reduced  to  the  same  form  are 
called  equivalent  equations,  or  dependent  equations. 


FIRST   DEGREE   EQUATIONS.  21 

66.  Equations  which  can  be  reduced  to  such  form  that  the 
coefficients  of  the  unknowns  are  the  same,  while  the  known 
quantities  are  different,  are  inconsistent  equations. 

67.  If  we  have  n  equations  containing  n  unknowns,  the 
equations  can  in  general  be  solved.  When  solution  is  impos- 
sible, there  will  occur  somewhere  in  the  work  either  equivalent 
or  inconsistent  equations.  Neither  equivalent  nor  inconsistent 
equations  can  be  solved. 


CHAPTER  VTII. 

INVOLUTION  AND   EVOLUTION. 

MONOMIALS. 

It  is  assumed  that  m  and  n  are  positive  integers. 

68.  The  92th  root  of  a  is  a  quantity  which,  raised  to  the  Tith 
power,  produces  a. 

Or,  icTaf^a. 

69.  It  is  evident  from  multiplication  that : 

a^'^O'  =  a'»+",  [1] 

(a'^y  =  a"*",  [2] 

(ab)"^  =:  drh^.  [3] 

These  are  fundamental  equations. 

It  is  also  evident  that : 


VCa"*")  -  V(a"r  =■  ^^  (4) 
It  will  now  be  shown  that  the  mth  power  of  the  wth  root  is 
equal  to  the  nth  root  of  the  mth  power  ;  or,  that : 

(Va)'»-:V^.  (5) 

Let                                     ^~a  =:  I  (6) 

Then                              (Va)"*-/"*;  (7) 

also,  from  (6),                           a  =  Z",  (8) 

or,                                              a'"  =  Z"*^ ;  (9) 

therefore,                              -y'a^  =  l^  =  (Va)"*,  (10) 
which  gives  (5)  as  required. 


INVOLUTION   AND   EVOLUTION.  23 

70.  VaVb  =  ^^,  [4] 
To  prove  this,  from  68  and  [3], 

{VaVbf^ab-, 

therefore,  Va  VJ  =  Va?. 

Also,  y^^VVa  =  -Va.  [5] 

To  prove  this,  let  Vv^  ==  ?;  (1) 

then  a  =  (Z^)"^=Z^'*;  (2) 

therefore,  I  -  V  V^  =  VV^  :=  "V^.  [5] 

Also,  Va  Va  =  "Va"*+^  [6] 

To  prove  this :  Va  —  "Va^  (7) 

and  Va-  v^^;  (8) 

therefore,  Va  Va  —  "Va"'"^^  [6] 

71.  A  fractional  exponent  is  explained  as  meaning,  by  its 
numerator  the  power  to  which  the  quantity  is  to  be  raised, 
by  its  denominator  the  root  which  is  to  be  taken.     Thus : 

ai=  Vci?^  {Vaf, 

This  explanation  will  be  seen  to  be  consistent  with  the  use  of 
integral  exponents. 

72.  Formulas  [6],  [5]  and  [4]  mean  that  [1],  [2]  and  [3] 
are  true  when  vi  and  n  are  positive  fractions  whose  numera- 
tors are  1  ;  and  by  a  simple  extension  of  70  it  is  seen  that  [1], 
[2]  and  [3]  are  true  for  all  positive  values. 

73.  Negative  exponents  may  be  viewed  as  occurring  thus  : 
Divide  a?  successively  by  increasing  powers  of  a  : 

a 


24  PEINCIPLES   OF    ELEMENTARY   ALGEBRA. 


-z  ~  0?'^  =  a^,  and  as  -^  ~  1 

we  shall  explain  a^  as  equal  to  1. 

—  =:  a^  *  =  a  ^  and  as  -^  =  - 
a*  a*      a 

we  shall  explain  a~^  as  equal  to  -• 

74.  The  reciprocal  of  a  quantity  is  1  divided  by  that 
quantity.  A  negative  exponent  means  the  reciprocal  of  the 
quantity  with  an  equal  positive  exponent. 

a**  a"' 

It  can  be  shown  that  [1],  [2]  and  [3]  are  true  when  m  and  n 
are  negative  ;  it  follows  that  they  are  true  universally. 

75.  A  radical  quantity  is  the  indicated  root  of  some  quan- 
tity. When  the  root  cannot  be  extracted  the .  indicated  root 
is  sometimes  called  a  surd. 

76.  A  radical  is  said  to  be  in  its  simplest  form  w^hen  every 
possible  operation  indicated  by  the  root  index  (or  by  the 
denominator  of  the  fractional  exponent)  has  been  performed. 
A  few  of  the  more  common  reductions  are  illustrated  in  these 
examples : 


(1)  Va^  +  aV  ==  V^Vl  +  aV  ==:  aVl  +  aV. 


(2)  mV2  +7n=  Vm' -V2  +  7n=  -</'2m'  +  7m'n. 

(3)./7^^^V7V2=.:^  =  lVl4. 
^^      V2      V2V2         2         2 

^  ^    ^2      ^/9.V9?        2        2 


INVOLUTION   AND    EVOLUTION.  25 


^  ^    \2-x  2-x  2~x 


■x\ 


(6)  V2+ V^_(V2+ V^.)(V2  +  V^)_2  +  2V2:r  +  ^ 
V2  -  V^      ( V2  -  V^)(  V2  +  V5)  2  -  :^ 

In  (3),  (4),  (5)  and  (6)  the  denominator  is  cleared  of  radi- 
icals,  often  a  desirable  result. 

77.  Radicals  can  be  combined  by  addition  and  subtraction 
only  when  the  radical  parts  consist  of  the  same  quantity  under 
the  same  index ;  such  radicals  are  called  similar  radicals. 
Thus  : 

(1)  V8+ V32  =  V?2+ VT6^-2V2  +  4V2  =  6V2.. 

(2)  V8+  v^8- V2-2V2  +  2-V2  =  2  + V2; 

2+V2=  \/2(a/2^+1). 

78.  Radicals  can  be  brought  under  one  sign  in  multiplica- 
tion and  division  only  when  the  factors  have  the  same  index. 
They  can  always  be  reduced  to  the  same  index.     Thus : 

POLYNOMIALS. 

79.  If  we  raise  (a  + J)  to  successive  powers,  we  shall  find 
certain  laws  governing  the  number  of  the  terms,  the  expo- 
nents of  the  letters,  and  the  coefficients.     Thus  : 

{a  +  6)^  =  a^  +  3  a^h  -\-ZaV-\-  h\ 

(a  +  by  =  a'  +  4:a'b  +  6a'P  +  4:aP  +  h\ 

{a-bf  =  o^  -  ba'b+lOa'b''  -lOa'b^  +  bah'  -bK 

The  law  of  the  number  of  terms  and  of  the  signs  and  the 
law  of  the  exponents  are  obvious ;  and  the  symmetrical 
arrangement  of  the  coefficients  is  noticeable. 


26  PRINCIPLES   OF   ELEMENTARY   ALGEBRA. 

The  law  of  the  coefficieDts  is  : 

If  we  multiply  any  coefficient  by  the  exponent  of  a  in  that 
term,  and  divide  the  product  by  a  number  one  greater  than 
the  exponent  of  h  in  that  term,  we  obtain  the  next  coefficient. 

80.  (a  +  5  +  c?+ 7  =  a'+2a5  +  i'  +  2ac  +  26c+c'  + 

81.  By  means  of  the  formula, 

the  square  root  of  a  polynomial  may  be  found.     Thus, 

a'  +  2aZ)  +  5^  =  a'  +  (2a  +  h)h. 

After  the  term  corresponding  to  a  has  been  found,  we  find 
the  next  term  by  dividing  the  "2a6  term"  by  "2a."  After 
subtracting  "  (a  +  ^)*^ "  the  "  (a  +  5)  "  is  treated  as  one  quantity 
(say  ^),  and  another  term^in  the  root  is  found  by  the  same 
formula. 

82.  By  means  of  the  formula, 

(a  +  Vf  =3  a^  +  3a^S  +  3a5^  +  h^ 
-:a^+(3a'  +  3aS  +  5')5, 

the  cube  root  of  a  polynomial  may  be  found.  The  process  is 
similar  to  the  extraction  of  square  root,  and  by  similar  for- 
mulas for  higher  powers  the  corresponding  roots  may  be 
obtained. 


CHAPTEE  IX. 

EATIO   AND   PROPORTION. 

83.  The  ratio  of  two  numbers  is  the  quotient  of  the  first 
divided  by  the  second.  The  division  is  indicated  by  a 
colon.     Thus : 

7       a 
a\  0  =  -. 

0 

84.  The  first  quantity  (or  the  first  term)  (a)  is  called  the 
antecedent,  the  second  the  consequent. 

85.  An  inverse  ratio  is  the  reciprocal  of  the  direct  ratio. 
Thus,  if  a  :  6  is  assumed  as  a  ratio  it  is  called  the  direct  ratio 
of  a  to  5,  and  5  :  a  is  the  inverse  ratio. 

86.  A  proportion  is  an  equality  of  ratios. 

87.  The  first  and  fourth  terms  are  called  extremes,  the 
second  and  third  means.  Thus :  a\b=^7)i:n  (also  written 
a  :  6  :  :  m  :  n)  is  a  proportion  ;  a  and  ?2  are  extremes,  b  and  m 
means. 

88.  We  have  a  \  h  w  m  :  n^  (1) 
to  prove                                a:m  :  \b  \n,  (2) 

From(l),  ^  =  -.  (3) 


a  : 

5:: 

m 

:n, 

a  : 

m  : 

•b 

:  n. 

a 
V 

_m 
n 

a 

_b 

m 

n 

a  : 

TYi : 

:b: 

:  n. 

Multiplying  by  1,        ^  =  "-,  (4) 

or,  a\m:  \b  \n,  (2) 


28  PRINCIPLES   OF   ELEMENTARY   ALGEBRA. 

89.  We  have  a  :  b  \ :  m  :  n,  (1) 
to  prove                        h '.  a  \\  n  \  m,  (2) 

From(l),  ^  =  ^.  (3) 

on 

Divide  1  by  both  members, 

.^  =  ^,  '  (4) 

a     m 

or,  h  \a\  \n:m,  (2) 

90.  We  have 

a:b  \\m\n^  (1) 

to  prove  ak-\-hl'.ar-{'hs:\7nh-\~nl\7nT-\-ns.    (2) 

From(l),  f  =  ^.  (3) 

0      n 


Multiply  by  -,  add  1  and  reduce, 
s 

ar  -f-  bs onr  -j-ns 

bs  71S 

Divide  1  by  (4)  and  multiply  by  -» 

s 

bl  nl 


ar  +  bs      mr  -f  ns 
Divide  1  by  (3), 


b  __n 
a     m 


Multiply  by  -,  add  1  and  reduce, 


bs  -\-  ar  __  ns  +  mr 
ar  mr 


(4) 


(5) 


(6) 


(7) 


EATIO   AND   PEOPOETIOJSr.  29 


Divide  1  by  (7)  and  multiply  by  -, 

r 


lea  hm  .Q. 


bs  -\-  ar  ns  -]-  mr 
Add  (5)  and  (8), 

ak~\-hl  mk-\-nl  /n\ 

ar  +  bs  rar  +  ns 

or,  dk -^  bl  \  ar  -\- bs  w  mk  +  nl :  mr  +  ns.        (2) 

By  giving  special  values  {e.g.,  1  or  0)  to  one  or  more  of  the 
quantities  ^,  ^,  r  and  s,  other  formulas  may  be  obtained  from 
this. 


CHAPTER  X. 

QUADRATIC    EQUATIONS. 
WITH    ONE    UNKNOWN    QUANTITY. 

91.  A  quadratic  equation  is  one  which  contains  the  second 
power,  and  no  higher  power,  of  the  unknown  quantity. 

92.  A  pure  quadratic  is  one  which  contains  the  unknown 
quantity  only  in  the  second  degree. 

93.  An  affected  quadratic  is  one  which  contains  the  un- 
known in  both  the  first  and  second  degree. 

94.  A  pure  quadratic,  or  a  pure  equation  of  any  degree,  is 
solved  by  arranging  the  unknown  quantity  on  one  side  of  the 
equation,  and  the  known  quantities  on  the  other,  and  ex- 
tracting the  root  of  both  sides. 

A  pure  quadratic  can  be  reduced  to  the  form 

x'  =  a,  (1) 

which  gives  x  = -\-  Va  and  —  Va  ;  (2) 

the  two  values  of  x  are  often  abbreviated  into  dbVa,  where 
the  sign  dz  means  +  and  —.  Every  root  of  even  index  will 
involve  the  double  sign. 

95.  If  the  two  members  of  a  pure  quadratic  equation  have 
opposite  signs,  the  value  of  the  unknown  is  the  indicated 
square  root  of  a  negative  quantity.     The  indicated  even  root 


QUADRATIC   EQUATIONS.  31 

of  a  negative  quantity  is  called  an  imaginary  quantity,  and  is 
so  distinguished  from  the  real  quantities  we  have  so  far  con- 
sidered. 

If  x^  =  —  ly  then  X  =  ±  V~  I  =  dz  V7  V—  1,  an  imaginary. 

96.  Every  affected  (or  complete)  quadratic  may  be  arra^nged 
in  the  following  form,  called  the  general  equation  of  the  sec- 
ond degree  : 

x'^  +  cx-^-n  —  O,  (q) 

To  solve  this,  transpose  the  n, 

x^  +  ex  =  —  n,  (1) 

or,  x"^  +  2-x  =  —  n.  (2) 

Eeferring  to  45,  (1)  it  is  plain  we  can  make  the  first  mem- 
ber of  (2)  a  square  of  a  binomial  by  adding  ( -)  •  Adding 
this  we  get 


^^  .-   t    /^  \^  -    ^   (^ 


.■^+2-^.+(^|j  =  -.+^y.  (3) 


■4n  , 


ing  square 

root, 

a:  =  — 

1 

4 

=.# 

—  in 
4      ' 

iw^ 

-in 
4      ' 

(4) 

(5) 

(6) 
(7) 


32  PEINCIPLES   OF    ELEMENTARY    ALGEBEA. 


^  =  ^l-  c  +  Vc'  -4:71]  =  a,  (a) 

A 
a  and  /3  are  the  roots  of  the  equation. 

97.  It  is  plain  that 

a  +  ^  =  -C, 

and  af3  =  n; 

that  is,  the  sum  of  the  roots  is  equal  to  minus  the  coefficient 
of  X  in  the  general  equation,  and  the  product  of  the  roots  is 
equal  to  the  term  without  x. 

98.  From  (a)  and  (5)  we  get, 

a;  —  a  =  0,  (ai) 

x-l3  =  0.  (bO 
Multiply  (ai)  by  (6i),  and  we  get 

(x-aXx-/S)  =  0,  (1) 

or,                               x'  ~-(a  +  ^)x  +  al3  =  0,  (2) 

or,  x'^ -{-  ex -{- n  =  0 ;  (q) 
that  is,                     x'^  +  cx-{-n  =  (x  —  a)(x  — 13). 

Therefore  a  quadratic  expression  x'^  -}- ex  ~{- n  is  factorable 
into  (x  —  a)(x  —  /3)j  where  a  and  ^  are  the  roots  of  the  equa- 
tion x^  -\-  ex  -^  n  =  0. 

When  a  and  ^  are  small  integers,  it  is  easy  to  detect  these 
factors  by  inspection,  45,  (9),  and  thus  to  discover  the  roots  at 
once. 

99.  Any  equation  of  the  form 

^^**  +  ex""  -\-  n  =  0 
may  be  solved  in  a  similar  manner. 


QUADEATIC   EQUATIONS.  33 

100.  An  irrational  equation  is  one  in  which  the  unknown 
quantity  is  under  the  radical  sign.  The  solution  of  an  irra- 
tional equation  often  involves  some  method  of  reduction  illus- 
trated under  76,  or  some  similar  method.  Also  they  often 
involve  the  solution  of  a  quadratic. 

101.  Whether  a  and  /?  are  both  real  or  both  imaginary,  and 
if  they  are  both  real  whether  they  are  both  positive  or  both 
negative,  or  one  positive  and  one  negative,  depends  on  the 
relative  values  of  c  and  n. 


WITH   TWO   UNKNOWN   QUANTITIES. 

102.  There  are  certain  classes  of  simultaneous  equations 
where  one  or  both  the  equations  are  quadratic,  that  can  be 
solved  by  elementary  methods. 

First.  When  one  equation  is  of  the  first  degree,  one  of  the 
unknowns  may  be  written  in  terms  of  the  other  (as  x~ay-]-h), 
and  substituting  this  value  of  x  in  the  quadratic,  the  resulting 
equation  is  a  quadratic  with  one  unknown,  and  can  be  solved. 

Second,   When  the  equations  can  be  reduced  to  the  form 

x"  +  a^x  +  5iy  +  ^1  =-  0  I  .^. 

or  to 

xy  +  a^x  +  &i3/  +  Ci  ==  0 1  ^2) 

xy  +  a^x  +  b>jy  +  (?2  =  0  j 

the  second  degree  term  can  be  eliminated,  and  by  substituting 
as  in  the  first  case,  we  get  a  quadratic  with  one  unknown, 
which  can  be  solved. 

Third.  When  the  equations  are  of  such  form  that  they  may 
produce  an  equation  of  the  form 

x"  +  axy  +  by""  =  0,  (8) 


34  PRINCIPLES   OF    ELEMENTARY    ALGEBRA. 

we  can  obtain  a  value  for  x  in  terms  of  y  by  completing  the 
square  and  reducing ;  and  we  can  then  substitute  as  above. 
Equations  which  will  produce  (3)  are  of  the  form 

I  x"-  +  aixy  +  h^"^  +  CiX  —  0  (4) 

l  r?:^  +  <^2^y  +  i^'iV^  +  ^2^  =  0,  (5) 

^^^  j  ^'  +  «i^y  +  ^ly'  +  ^1  =  0  (6) 

1  x"^  +  a,^y  +  h^y''  +  ^2  =-  0.  (7) 

By  eliminating  the  first  degree  or  the  zero  degree  term,  the 
form  (3)  is  obtained.     Special  forms  of  (4)  are, 

(8) 

(9) 

(10) 

103.    There   are  special  devices  which  can  sometimes   be 
employed,  but  no  general  rule  for  them  can  be  given. 


xy  +  cx  =  0, 

or, 

ocy  -\-  c   =0, 

or, 

x"  ^c   =-0. 

CHAPTER  XI. 

PROGRESSIONS. 

104.  A  series  is  a  set  of  terms  which  succeed  each  other  by 
some  general  law. 

105.  An  Arithmetical  Progression  is  a  series  in  which  every 
term  is  equal  to  the  preceding  term,  plus  or  minus  some  fixed 
quantity.  This  fixed  quantity  is  called  the  common  differ- 
ence.    Thus  : 

a,   a-\-d,   a  +  2c? , 

-19,   -16,    -13 , 

3J,   3,    2J,   2 , 

are  Arithmetical  Progressions. 

106.  Let  a  —  the  first  term, 

Z  =  the  last  term, 
n  =  the  number  of  terms, 
d  —  the  common  diflference, 
5  —  the  sum  of  the  series  ; 

then  from  the  series  a,  a  +  d,  a+2d it  is  evident  that 

the  last  term  equals   a-{-  (n—l)d; 

i.e.,  l  =  a  +  (n—l)d.  [I] 

107.  s  =  a  +  (a  +  d)  +  (a  +  2d)  +  '""  +  (l-d)  +  l,  (1) 
inverting  the  order, 

s  =  l  +  (l.~.d)  +  (l-2d)  + +  (a  +  d)  +  a,      (2) 


36  PRINCIPLES    OF    ELEMENTARY    ALGEBRA. 


adding, 


2s-=a  +  l+a  +  l-{-a+l+ +  a+l+a-\-l.      (3) 

The  number  of  terms  in  (1)  and  (2)  is  ?z, 
therefore,  2  s  ==  n(a  +  Z),  (4) 

or,  s=    ^  ^   ^.  [s] 

108.  In  [I]  and  [s]  together  we  have  the  quantities  a,  I,  n, 
d,  and  s ;  we  can  eliminate  any  one,  and  so  express  any  one  in 
terms  of  any  other  three. 

109.  A  Geometrical  Progression  is  a  series  in  which  every 
term  bears  a  fixed  ratio  to  the  preceding  term.     Thus  : 

a,  ar,  ar^,  ar^ , 

-2,     6,  -18,     54 , 

12,  6,  3 , 

are  Geometrical  Progressions. 

110.  Let  r  denote  the  ratio,  and  use  a,  I,  n,  and  s  as  above. 
Then  in  the  series  a,  ar,  ar'^ ,  the  last  term  equals  aT'^~^, 

or,  l-=af-^,  [Z] 

111.  s  =  a-\-aT-\-ai^  + +ar^~'.  (1) 

Multiply  by  r, 

8r  ^=^  ar -\- ai^  ■\- ar^ -\- -^-ar"^,  (2) 

(2)  -  (1),  sr  -  5  -  af"  -  a,  (3) 

air""—!)  rcy-i 

or,  s  =  -^ -^-  [S] 

r—  1 

If  r  is  less  than  1,  [S]  is  more  conyemently  used  in  the  form 
1  —  r 


PEOGEESSIONS.  37 

112.  By  combining  [L]  and  [S],  other  formulas  may  be 
obtained. 

113.  If  r  is  a  proper  fraction,  the  quantity  r"  can  be  made 
as  small  absolutely  as  we  please  by  increasing  the  value  of  n. 
It  can  thus  be  made  to  approach  0  as  nearly  as  we  please,  but 
will  always  differ  slightly  from  0.  Such  a  quantity  is  said  to 
have  0  for  its  limit.  It  is  ;^lain  that  if  ar"'  approaches  0  as  a 
limit,  the  quantity 


a(l  — r^)  _a- 


ar'" 


1  —  r  1  —  r 

must  approach  for  its  limit      ^    .     That  is,  the  limit  of  the 

1  —  r 

sum  of  the  series  is when  n   is   indefinitely   increased. 

1  —  ?' 

This  is  sometimes  abbreviated  in  the  formula, 

where  =  means  ap^^roaches,  oo  means  infinity^  and  lim  means 
limit  of. 


APPENDIX. 
A. 

PROOF   OF   RULE   FOR   FINDING   THE   G.C.D. 

114.  The  principle  of  39  will  first  be  proved.  Let  Jf  and 
N  denote  the  given  polynomials,  q,  r  and  s  the  quotients, 
c  and  d  the  remainders.     Indicate  the  divisions  thus  : 

M)  N{q 
Mq 
c)M(r 
cr 

d)  c  (s 
ds 
~0 

We  will  first  show  that  d  is  a  divisor  of  M  and  iV". 

N=Mq  +  c,  M=cr  +  d,  c^ds\ 

then,         M=  dsr  +  d=  d(sr  +  1), 

and  iV=  (dsr  -{-  d)  q  -\-  ds  =^  d(srq  -{-  q-{-  s); 

therefore  d  divides  both  M  and  JSF.  We  will  next  show  that 
d  is  the  greatest  common  divisor  of  Jfand  iV". 

Every  divisor  of  If  and  iV  divides  JV—  Mq,  that  is,  c ; 
therefore  every  divisor  of  JIf  and  iVis  a  divisor  of  J/ and  c. 

Every  divisor  of  iHf  and  c  divides  M—  cr,  that  is  d]  there- 
fore every  divisor  of  J/and  (?  is  a  divisor  of  c  and  d. 

Therefore  every  divisor  of  M  and  iV  is  a  divisor  of  d. 
There  is  no  expression  higher  than  d  which  can  divide  d; 
therefore  d  is  the  greatest  common  divisor. 


APPENDIX.  39 

115.  In  regard  to  40,  if  all  the  monomial  factors  are  re- 
moved from  the  polynomials  before  division  begins,  it  is  plain 
that  the  G.C.D.  of  the  resulting  polynomials  must  contain  no 
monomial  factor.  Whatever  monomials  we  may  introduce  or 
remove  during  the  work  can  make  a  difference  only  in  the 
coefficients,  and  if  at  the  last  we  remove  all  monomial  factors 
from  the  last  divisor,  it  must  be  the  G.C.D.  of  the  polynomials 
with  which  we  begin.  We  are  therefore  justified  in  using  the 
suggestions  of  40. 

To  find  the  G.C.D.  of   * 
3:r^  —  10^^'  +  15:r  +  8  and  o;^  -  2:r*  -  6x'  +  4:x''  +  13:r  +  6. 

x'-2x'-~6x'+4:x''+13x  +  6)3x'  -10:r^  +lbx+  8(3 

Sx'-6x'-18x'+12x''+S9x+ 18 
6x'+  Sx'-l2x''-24:x-lO 

Divide  the  new  divisor  by  2  and  multiply  the  new  dividend 
by  3, 

3^*+4:r^-62^''^-12^-5)3;r^-  6;r^-18:r^  +  12:r^+39:i'+18(5; 
Sx'+  4:x'-   Qx'-Ux'-  bx 

~10:r^-125;^  +  24^'^-44:?;  +  18 

Divide  this  remainder  by  2,  multiply  it  by  3,  and  use  the 
same  divisor  again, 

-  \bx>  -  18^'  +  36;?:^  +  66a;  +  27(~  5 
-\bx^  —  2Q)x^  +  30^^  +  60.T  +  25 
2x^-\-    6.r''+    6a;  +    2 

Divide  the  new  divisor  by  2, 

^3  _^  3^2  j^^x^  Y)Zx'  +  4r'  -    ^x^  -  \2x  -  5  (3a;  -  5 
3^M-9^M-9^'+    3a; 


—  5  o;^  —  15  a;^  —  15  a;  —  5 

—  5a;^  —  15a;^  —  15a;  —  5 


a;^^  32?'  +  3a;+  1  is  the  G.C.D.  required. 


40  PEINCIPLES   OF    ELEMENTARY    ALGEBRA. 

B. 

ILLUSTRATIONS   OF   SQUARE   ROOT   AND    CUBE   ROOT. 

116.    To  find  the  square  root  of  4^^ -  12a;'  +  5x'  +  Qx+l. 

4.x'  -  12^'  +  bx''+6x+l  (2x'  -Sx-1 
4.x' 


4:r^-3a;)       -l^x'  +  bx" 
-I2x^  +  ^x' 

4:X^~^X-l)  —4:X^+^X+1 

—  4^'  +  6:r+l 

After  the  second  subtraction  we  have  subtracted  (2^^  —  Zxf. 
If  we  call  (2x^  —  2>x)=^  I,  the  second  trial  divisor  is  2/,  the 
second  complete  divisor  is  (21—  1);  in  the  third  subtraction 
we  subtract  —  1*(2Z— 1),  and  we  may  regard  the  polyno- 
mial as  represented  by  Z^  —  2  Z  +  1  ~  ^^  —  1 '  (2  /  —  1)  ;  which 
accords  with  the  formula  in  81. 

117.    i/x^-^x'  +  ^0x^-^^x~Q4:  =  '^ 
x^ 


Zx'-^x''-\-^x')~^x''  +^0x^ 

-6:2;^+12^'-  Sx^ 

3:r*-12^+12:z;^ 


-12a;^+24:r+16 


2>x'~l2x^  +24:z;+16 


-12:r^+48;r'~96^-64 
-12^*+48:i-'-96^-64 


We  may  here  regard  {oc^  --2x)=^l,  and  it  will  be  seen  that 
this  process  accords  with  the  formula  for  (a  +  hf  in  82. 


APPENDIX,  41 


ILLUSTRATIONS   OF   EXAMPLES   SOLVED   LIKE  QUADRATICS. 

118.    x  +  4cVx  =  21.     LetV^  =  3/,   then  x  =  y\ 

.       3/^  +  4y^2L  (1) 

y(=:  V^)  =  3  and  =  —  7.  (2) 

a;  =  9  and  =-  49.  (3) 


119.   a;+V5^+l0  =  8. 


V5^  +  10-:8-a:,  (1) 

5x  +  10  =  (8-xy  =  64:-16x  +  x\  (2) 

a;  =  18  and  =  3.  (3) 

If  we  substitute  these  values  of  x  in  the  given  equation,  we 
shall  find  that  18  is  not  a  root  of  the  equation.  18  is  a  root 
of  the  equation  x  —  V5a;  +  10  —  8,  and  we  should  remember 
that  the  sign  ^  strictly  signifies  =b  V-  ^^  ^^  ^^^P  ^  P^^^  ^^ 
the  symbol  dz,  and  solve  the  equation  by  squaring,  we  must 
test  the  answers  to  be  sure  that  we  have  true  roots. 


120.  x''-7x+Vx''-7x+lS  =  24,  (1) 

add  18 to  (1),  o;^ -  7:r  +  18  +  Va;'-7^+18  =  42.  (2) 


(3) 

(4) 
Ans.  x  =  9;  x  =  —  2.. 


Call 

y  = 

--Vx' 

-7: 

r+18, 

f 

+  y 

=  42, 

y- 

=  6 

and  = 

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